Consider a non-conducting plate of radius r and mass m which has a charge q distributed uniformly over it. The plate is rotated about its axis with an angular speed ω. Show that the magnetic moment μ and the angular momentum ℓ of the plate are related as

Given-
Radius of the ring = r

Mass of the ring = m

Total charge of the ring = q

Angular speed,

Where

F is the frequency

Now frequency f

Where,

T is the time period

Magnetic moment,

where,

A= area of cross section

i=current flowing through it

n = number of turns

for number of turns n = 1

Current in the ring,

For the ring of area A with current i, magnetic moment,m is given by

Where A is the area of the loop

Angular momentum,

where

I is the moment of inertia of the ring about its axis of rotation.

is the angular moment

Moment of inertia, I is given by-

where,

m is the mass of the object

r is the radius of the circular object

So, substituting in (2)

Putting this value in equation (1),